CERN-TH/99-379 THES-TP/99-xx hep-ph/9912253 December 1999 Loop-Induced CP Violation in the Gaugino and Higgsino Sectors of Supersymmetric Theories Apostolos Pilaftsis Theory Division, CERN, CH-1211 Geneva 23, Switzerland and Department of Theoretical Physics, University of Thessaloniki, GR 54006 Thessaloniki, Greece ABSTRACT We show that the gaugino and higgsino sectors of supersymmetric theories can nat- urally acquire observable CP violation through radiative effects which originate from large CP-violating trilinear couplings of the Higgs bosons to the third-generation scalar quarks. These CP-violating loop effects are not attainable by evolving the supersymmetric renormalization-group equations from a higher unification scale down to the electroweak one. We briefly discuss the phenomenological consequences of such a scenario, and as an example, calculate the two-loop contribution to the neutron electric dipole moment gener- ated by the one-loop chromo-electric dipole moment of the gluino. PACS numbers: 11.30.Er, 14.80.Er 1 Supersymmetric (SUSY) theories, including the minimal supersymmetric Standard Model (MSSM), predict several new unsuppressed CP-violating phases which generally lead to too large contributions to the electric dipole moments (EDM’s) of the neutron and electron [1, 2, 3, 4, 5, 6]. Several suggestions have been made in the literature to overcome such a CP crisis in SUSY theories. Apart from the cancellation mechanism proposed in [5], an interesting solution to the above CP-crisis problem is to assume that the first two generations are either very heavy above the TeV scale [4, 7] or they do not involve CP- violating phases in their trilinear couplings Af to the Higgs bosons [6]. In the latter SUSY framework with all scalar quarks much below the TeV scale, we also have to require that the SU(3)c, SU(2)L and U(1)Y gaugino masses: mg~, mW~ and mB~ , as well as the higgsino-mass term µ be real parameters. Then, for both of the above scenarios, only the scalar-top (t˜) and bottom quarks (˜b) may be considered as potential mediators of CP non-conservation. Here, we shall adopt this minimal model of SUSY CP violation. We will not address the issue concerning the underlying mechanism of SUSY breaking that leads to the above low-energy scenario, as it is beyond the scope of the present study (see also [7]). In the above discussion, the CP-violating phases were defined in the weak basis in 2 which the soft-SUSY-breaking parameter m12 of Higgs mixing is real. At the tree level, such a convention leads to positive vacuum expectation values v1 and v2 for the two Higgs doublets Φ1 and Φ2 of the MSSM, where the relative phase ξ between v1 and v2 vanishes. If this phase convention is to be adopted order by order in perturbation theory [8], then the formally radiatively-induced phase ξ can always be eliminated by an appropriate choice of 2 the renormalization counter-term (CT) Im m12. This latter CT is related to CP-odd tadpole graphs of the pseudoscalar Higgs bosons, and is important to render the CP-violating Higgs scalar-pseudoscalar transitions ultra-violet finite [8] in Higgs-boson mass spectrum analyses 2 2 [9, 10]. However, it should be stressed that unlike Re m12,Imm12 does not directly enter the renormalization of the physical kinematic parameters of the MSSM, such as the charged Higgs-boson mass and tan β = v2/v1. Consequently, no additional CP violation can be generated through the phase ξ in the effective chargino and neutralino mass matrices. In this paper, we shall study a novel mechanism of radiatively inducing CP violation in the gaugino and higgsino sectors of the MSSM. In these sectors, CP violation is commu- nicated by trilinear interactions of scalar top and bottom quarks to the Higgs bosons. It is interesting to remark that this loop-induced CP violation cannot be achieved by running the renormalization-group equations from a higher unification scale down to the electroweak energies. Even though the discussion is confined to the MSSM, the results of the analysis can straightforwardly be generalized to other SUSY extensions of the SM. We start our discussion by considering the gluino sector. In the convention in which 2 Φ2 Φ† h i h 2i t˜ t˜ t˜ t˜ L × R L × R C C g˜R(p) tL × tR (˜gR) g˜R tL × tR (˜gR) (a) (b) Figure 1: Chromo-electric dipole moment of the gluino; attaching the gluon field to the top- and scalar-top- quark lines is implied. Φ h 2i q˜ q˜ L × R qL g˜R g˜L qR gµ(k) Figure 2: Two-loop CEDM of a light quark induced by the one-loop CEDM of the gluino. the gluino mass is real, i.e. arg (mg~) = 0, CP violation is induced by the interaction La- grangian: a a g~qq~ a λ a λ = √2gs q˜∗ g¯˜ PLq q˜∗ g¯˜ PRq +h.c., (1) Lint − L 2 − R 2 qX=t;b where PL R =(1 (+)γ )/2 is the left- (right-) handed chirality projection operator and ( ) − 5 λa, with a =1,...,8, are the usual SU(3) Gell-Mann matrices. In fact, CP violation enters through the mixing of the weak (˜qL, q˜R) with mass eigenstates (˜q1, q˜2). Such a mixing of states is described by the unitary transformation: q˜ q˜ 10 cos θ sin θ q˜ L = U q 1 = q q 1 , (2) iδq 0 q˜R 1 0 q˜ 1 0 0e1 0 sin θq cos θq 1 0 q˜ 1 2 − 2 @ A @ A @ A @ A @ A where δq =arg(Aq µ∗Rq), with Rt =cotβand Rb =tanβ, is a non-trivial CP-violating − phase. The gluino mass mg~ receives a finite radiative shift from diagrams depicted in Fig. 1, which can easily be calculated to be 1 2 2 αs mt x + M~ (1 x) 2 iδt t1 − δmg~ = mt sin θt cos θt e− dx ln 2 2 . (3) 4π mt x + M~ (1 x) Z0 t2 − 3 It is obvious that only scalar-top quarks are of relevance, as the effect of the scalar bot- 2 2 tom quarks is suppressed by a mb /mt -factor. For relatively large values of the scalar- top mixing, i.e. Mt~2 /Mt~1 5, θt 45◦, we find a radiatively-induced phase for mg~: 2 ∼ ≈ arg(mg) 10− mt/mg. Even though arg(mg) may formally be re-absorbed into the renor- ~ ≈ ~ ~ malization of mg~, its presence, however, will manifest itself in related CP-violating higher- point correlation functions,∗ such as the chromo-electric dipole moment (CEDM) form g~ 1 g~ abc¯b µν c a factor of the gluino described by the effective Lagrangian CEDM = 2 d f g˜ iσ γ5g˜ Fµν , a L where Fµν is the gluon field strength tensor and g~ 2 1 d (p ) αs t t x = m Im (U ∗U ) dx . (4) t 1i 2i 2 2 2 gs 4π mt x + M~ (1 x) p x(1 x) i=1X;2 Z0 ti − − − Notice that gluino being a Majorana particle with internal degrees of freedom, i.e. colour, can only possess a non-vanishing CEDM; it cannot have an EDM. Furthermore, at two loops, the CEDM form factor of Eq. (4) can in turn give rise to a CEDM of a light quark C C dq , as shown in Fig. 2. The contribution to CEDM of a light quark dq has been calculated to give C 2 M2 2 M2 dq 3αs mt t t q q q~j mt t~ = Im (U U )Re(U ∗U )H , , i , (5) 2 2 1i∗ 2i 1j 2j 2 2 2 gs 16π mg~ i;j ; mg~ mg~ mg~ X=1 2 where the two-loop function H(a, b, c) is defined as 1 dx x 1 a a ln a H(a, b, c)= − + 0 (1 a)2 x(1 x) bx c(1 x) ax(1 x) bx c(1 x) Z − − − − − − − − − (1 a)2 [bx + c(1 x)] x(1 x) bx + c(1 x) − − − ln − .(6) − [ax(1 x) bx c(1 x)][x(1 x) bx c(1 x)]2 x(1 x) − − − − − − − − − Based on the naive quark model and after including QCD renormalization effects, the contribution to the neutron EDM may be estimated by dC g (M ) 120=23 4 dC 2 dC n s Z d + u , (7) e ≈ gs(Λ) 9 gs Λ 9 gs Λ C C where gs and the u-andd-quark masses in du and dd are calculated at the scale Λ: mu(Λ) = 7 MeV, md(Λ) = 10 MeV, and gs(Λ)/(4π)=1/√6 [3]. Numerical estimates C of dn /e are exhibited in Table 1, for a non-universal scenario for the scalar-quark masses with moduli-universal trilinear couplings. We assume that the first two generations possess ¯ ∗As was discussed in [2], arg(mg~) may also contribute to the CP-violating phase of QCD θ.Herewe shall not deal with this strong CP problem, but merely assume that θ¯ is eliminated by a chiral rotation of the quark fields. 4 C 26 C 26 tan β Mu;d d /e [10− cm ] d /e [10− cm ] | n | | n | [GeV] m = 200 GeV m = 400 GeV f g~ g~ µ = 0.5, 1, 2 TeV µ = 0.5, 1, 2 TeV 150 0.3 1.0 5.0 0.2 0.6 3.0 2 300 0.1 0.2 1.1 0.1 0.2 0.9 150 1.3 4.2 10.